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Báo cáo toán học: "Balancing cyclic R-ary Gray codes II"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Balancing cyclic R-ary Gray codes II. | Balancing cyclic R-ary Gray codes II Mary Flahive Department of Mathematics Oregon State University Corvallis OR 97331 USA flahive@math.oregonstate.edu Submitted Jan 15 2008 Accepted Oct 3 2008 Published Oct 13 2008 Mathematics Subject Classification 05A99 05C45 68R10 Abstract New cyclic n-digit Gray codes are constructed over 0 1 . R 1 for all R 2 n 3. These codes have the property that the distribution of digit changes transition counts between two successive elements is close to uniform. For R 2 the construction and proof are simpler than earlier balanced cyclic binary Gray codes. For R 3 and n 2 every transition count is within 2 of the average Rn n. For even R 2 the codes are as close to uniform as possible except when there are two anomalous transition counts for R 2 mod 4 and Rn is divisible by n. 1 Introduction For fixed integers R n 2 an n-digit R-ary Gray code is an ordering of all n-strings with digits from 0 1 . R 1 g such that any two consecutive strings differ in only one digit and that difference is 1. When the last and first strings also satisfy this property the code is called cyclic. For example 20 21 22 12 02 01 11 10 00 1.1 is a cyclic 2-digit ternary Gray code. We only consider cyclic codes here. In the mid-twentieth century Frank Gray 7 designed the eponymous Binary Reflected Gray Code to facilitate relaying information through many repeaters. His code can be generalized to R 3 and is cyclic when R is even. The term Gray code is now often used to describe listings of combinatorial objects in which successive elements differ in some prescribed minimal way cf. 11 but here the term is not used in that general sense. The transition sequence of a cyclic Gray code records the succession of digit changes d1 . dRn in the code the number of times a digit occurs in the transition sequence is THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R128 1 called its transition count and the collection of transition counts is the transition spectrum of the code. .